R-Ai63 989 THE DETUNSLING OF AN AXIRLLY SYMMETRIC SATELLITE WIITH jijlAN ORBITAL MANEUVER (U) AIR FORCE INST OF TECHNRIGHT-PATTERSON AFB OH SCHOOL OF ENGI K R FLEMING

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Si LECTE fI FEB 13 M

E DETUMBLING OF AN AI SMRC

I SATELLITE WITH AN ORBITAL MANEUVERINGRVEHICLE BY NONLINEAR FEEDBACK CONTROL

*~ I THESIS

" I Kir R. Flemin

I Appioved o public r-- -- o-,LUJ Distribution Unfird.tod

C.D DEPARTMENT OF THE AIR FORCE

AIR UNIVERSITY

SAIR FORCE INSTITUTE OF TECHNOLOGY

I Wright- Patterson Air Force Base, Ohio

I I 1

-- aAFIT/GA/AA/85D-5

I

DTICELECTE 0

SFEB 1 338

THE DETUMBLING OF AN AXIALLY SYMMETRIC

SATELLITE WITH AN ORBITAL MANEUVERING

VEHICLE BY NONLINEAR FEEDBACK CONTROL

THESIS

Kirk R. FlemingFirst Lieutenant, USAF

LAFIT/GA/AA/85D-5

L

,I.', .. Approved for public release; distribution unlimited

L

AFIT/GA/AA/85D-5

I

THE DETUMBLING OF AN AXIALLY SYMMETRIC SATELLITE

WITH AN ORBITAL MANEUVERING VEHICLE

BY NONLINEAR FEEDBACK CONTROL

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

In Partial Fulfillment of the

Requirements for the Degree of

Master of Science in Astronautical Engineering

P Accesion For

NTIS CRA&IDTIC TAB

UnannoutcedJustificatiol,

Kirk R. Fleming, B.S..Kirk~B ... ....ng B.S..........'..... ............. .

First Lieutenant, USAF Distrit'tio.

Availability Codes

Dist pIalj/ December 1985 Special

Ai-1Approved for public release; distribution unlimited

If

Table of Contents

Page

Acknowledgements ....... .. . ............... ii

List of Figures .... . .. .. .. ............... iii

List of Tables ......................... v

Abstract ....... . .. . ............. ..... vi

Notation ......... ................. vii

I. Introduction ...... . .. . . .............. 1

II. Problem Formulation ....... .. . ......... .. 4

System Configuration.. ....... ... 4The Hooker-Margulies Equations . . .. . .. . . 4Application of the HM Equations ...... . ...... 8

III. Nonlinear Feedback Control .. . ...... . . . ..13

IV. Results ....... .. ................ 20

V. Conclusion ....... ................. 46

References ........ ................. . 48

Vita .......... ................... 49

V

*w- . ' . , . , .,, . ..' ,, . , .. .:.L , .

Acknowledgments

I want to thank my thesis advisor, Lt. Col. Joseph W. Widhalm,

for his guidance during the course of this project. I also owe special

thanks to my classmates Robert Bandstra, William Berrier and Randall

Richey for their helpful insight and comments, particularly with regard

to my source code debugging efforts. I thank all my classmates for the

support they gave me in this last, very busy quarter.

,.

ii

List of Figures

Figure Page

1. The Conway and Widhalm Model for the Two Body Satellite . . 2

2. The Five-Body Satellite Model ................. 5

3. OMV Thrust Torques, Momentum Wheels Uncoupled . .... .. 26

4. Universal Joint Torques, Uncoupled Momentum Wheels ..... . 26

L 5. Joint Position and Precession Angle Decay .. ........ 27

6. Target Spin and Precession Angle Rate Decay . ...... . 27

7. Constraint Loads at Universal Joint ... ......... 28

8. Angular Velocities of OMV with u7 and u8 Feedback Only . . . 29

9. Angular Velocity of OMV with u7 and u8 Feedback Only . . . 29

10. Joint Constraint Loads, u7 and u8 Feedback Only .. ...... . 30

11. Control Torque History, u7 and u8 Feedback Only ..... .. 30

12. Target Spin and Precession Angle Rates, u7 and u8 Feedback Only 31

13. Target Precession Angle and Joint Position,u7 and u8 Feedback Only ............... .. 31

14. OMV Angular Velocity Components; Full Feedback Addedat t = 250 seconds .................... 32

15. OMV b3 Angular Velocity Component; Full Feedback AddedL at t = 250 seconds .................... 32

16. Precession Angle Rate :f Change and Target Spin Rate;Full Feedback Added at t = 250 seconds .. ........ 33

17. Gimbal Control Torque u7 With Full Feedback Addedat t = 250 seconds ................... . 33

18. Control Torque u and Constraint Torque TC; Full FeedbackAdded at t = 258 seconds ................. 34

19. Constraint Force Magnitude and 62 Component; Full FeedbackAdded at t = 250 seconds .............. . 34

iii

UList of Figures, cont'd

Figure Page

20. OMV Thruster Torques, b Momentum Wheel Torque Coupledto Target Precession Anle ................. 35

21. b2 Momentum Wheel Control Torque, u5, and WheelAngular Velocity; Control Torque u5 Coupled

to Target Precession Angle ................. 35

iv

List of Tables

I Table Page

I. Satellite Mass Properties .. ............ 22

II. Initial Conditions ................. 22

III. Gimbal Control and Constraint Torques,Uncoupled Momentum Wheels .. ........... .. 36

IV. Gimbal Control and Constraint Torques,B2 Momentum Wheel Coupled ........ ..... 38

V. Momentum Wheel and Thruster Control Torques,62 Momentum Wheel Coupled .. ......... .... 40

VI. Gimbal Control and Constraint Torques,Gimbal Torque Feedback Only Until t = 250 seconds . 42

VII. OMV Angular Velocity, Target Spin and Precession Angle Rate,1! Gimbal Torque Feedback Only Until t = 250 seconds . . . 44

P

v

EV

AFIT/GA/AA/85D-5

Abstract

The problem of detubling a freely spinning and precessing axi-

symmetric satellite is considered. Detumbling is achieved with another

axisymmetric orbital maneuvering vehicle (OMV) joined to the target sat-

ellite with a universal joint. The joint provides two rotational degrees

of freedom and is translated across the surface of the OMV during the de-

tumbling process. The target satellite and the OMV with its three mo-

mentum wheels are modelled as a five body system using Eulerian-based

equations of motion developed by Hooker and Margulies. A Liapunov tech-

nique is applied to derive a nonlinear feedback control law which drives

the system asymtotically to a final spin-stabilized state. State and

control histories are presented and indicate that the dtumbling process

is benign. Constraint force and moment loads at the connection between

ithe OMV and target satellites are also presented, and indicate that no

extreme loads are encountered during the despinning and detumbling process.

IA

vi

CI A

NotationBmk = mass of body A

m = total system mass

O = inertia dyadic of body A about its mass center

= angular velocity of body XF = non-gravitational external force on bodyTA = non-gravitational external torque on body A

iP = geocentric position vector for mass center of body A

F X" = interaction force acting on body A through joint jAj

= constraint torque acting on body k through joint jAL1 = unit dyadic

Y gravitational constant

= geocentric position vector for system center of mass

b p = unit vector in direction of

TSD = spring-damper torque acting on body X at joint jkj

gi = unit vector along rotation axis of joint i

Yi = angle of rotation about axis gi

(a 0 = angular velocity of the reference body (the OMV)

viiL- '

A.

THE DETUMBLING OF AN AXIALLY SYMMETRIC SATELLITE WITH AN

ORBITAL MANEUVERING VEHICLE BY NONLINEAR FEEDBACK CONTROL

I. Introduction

*The service or repair of orbiting satellites beyond direct reach

of the Space Shuttle may require an orbital maneuvering vehicle (OMV) to

C. rendezvous and dock with the target satellites. If the target 4s spin-

stabilized it may be necessary to despin it. If the target has experi-

enced control system malfunction or for some reason is not in pure spin,

it may be ecessary to detumble it. Docking followed by despinning or

detumbling is defined here as capture. Docking is accomplished bv first

driving a grappling device on the OMV to a state of rest relative to some

docking point on the target. The OMV and target can then be connected.

Despinning or detumbling is accomplished by applying torques to the tar-

get through the connecting joint while firing the OMV thrusters to control

the absolute motion of the two-body system. Widhalm and Conway derived a

Rfeedback control law (1) which solved the despinning/detumbling problem

for the case of axisymietric target and OMV satellites. They used a con-

necting joint which could translate across the surface of the OMV. The

translational degree of freedom of the joint is depicted in Fig. 1 by the

double arrow. The ability to translate the joint provides for joint pos-

ition adjustment during docking, and allows the joint to be driven to the

OMV axis of symmetry during detumbling. The resulting configuration can

then be spin-stabilized.

This thesis extends the Widhalm and Conway model to include three

orthogonal momentum wheels on the OMV, and develops a feedback control

.4

Body 1

Fig.1. heowy an0 ihi h woBd aelt

2

law to couple the momentum wheel torques to the system state. The target

satellite docked with the OMV and its three momentum wheels are modelled

as a system of five rigid, constant mass bodies. The control problem is

formulated by defining the system initial configuration and the desired

final state, and by deriving the equations of motion.

:.3p:

°53

N.,

II. Problem Formulation

System Configuration

An axisymmetric target satellite is docked with an OMV which,

with its three orthogonal momentum wheels, is also axisymmetric (see

Fig. 2). The target and OMV are connected with a universal joint having

two rotational degrees of freedom and the capability of translation

across the surface of the OMV. The translational degree of freedom is

depicted in Fig. 2 by the bold double arrow. The center of mass of the

target lies on the b3 axis as does the mass center of the OMV-mori ntum

wheel combination. The OMV is in a state of pure spin about b3, and the

target is in a state of spin with precession about 63 at a rate equal to

the OMV spin rate. With no external moments or forces acting on the sys-

tem a dynamically stable configuration results. This configuration rep-

resents the initial state of the system. The detumbling and despinning

process is complete when the joint has been driven to a position lying on

the 63 axis, and the target spin rate relative to the OMV is zero with

the OMV itself still in a state of pure spin about the 63 axis. This

configuration, or one arbitrarily close to it, represents the desired

final state of the system. Both the initial and final states as defined

imply that the initial and final angular velocities of the momentum

wheels having rotational freedom about the b and b axes are zero. The

initial and final angular velocity of the b3 wheel is arbitrary.

The Hooker-Margulies Equations

The dynamical attitude equations for a two-body satellite were

derived by Fletcher, Rongved and Yu (2), and were generalized for an

4 4

Bodyy 1

,e2 L3 /*

F. e e S i M

"" b3 b

Body 0 e3/

" ' Body 1 l

• Note:

"" I l [I61 i[ l i 4

-2,t'Fig. 2. The Five-Body Satellite Model

4,.

44 5 .

complete derivation is a set of 3n scalar equations for an n-body system.

The equations are free of the unknown joint constraint forces, but still

contain the unknown constraint torques (3:125).

* Hooker (4) showed in a subsequent paper that the constraint

torques could also be explicitly eliminated. The equations derived in

reference (3) are written for all the bodies iying on oite side of a se-

9lected joint and subsequently added. The interaction torques all cancel

in pairs, with the exception of the constraint torque at the free joint.

If the selected joint has a rotational degree of freedom about an axis

g, the dot product of g and the expression just found for the torque is

zero. Writing the dot product and setting it to zero yields an equa-

tion free of the constraint torque. Repeating the process for each de-

gree of freedom at each joint eliminates all the unknown constraint

torques and yields a system of r equations for an n-body system having

r rotational degrees of freedom. These equations are referred to in

some of the references as the modified Hooker-Margulies equations.

Although for an identical dynamical system a Lagrangian deriva-

tion would provide the same number of equations as the HM equations, the

resulting expressions would not be written in terms of the physical body

axes, as are the HM and modified HM equations. As a result adaptation

to active control and modification to include effects such as joint motion

would be more difficult (4:1205). The modified HM equations significant-

ly reduce the required computer time for solution due to the reduction

in the number of equations from 3n to r.

The constraint torques at the joints were eliminated explicitly

" by taking the dot product of the unit vector about which the joint is

* '6

'woo. 0

n-body system by Hooker and Margulies (3). Both derivations assume that

the bodies are connected by joints which are fixed with respect to the

bodies they connect. In addition, the generalized Hooker-Margulies (HM)

equations assume Lhat the bodies are -onnected in a topological tree.

This means that there are no closed loops formed by the interconnected

bodies. The restriction of immoveable joint is removed in an extension

of the HM equations which will be discussed later. The extended equa-

tions then apply to the OMV-target satellite system when modelled as a

system of five interconnected rigid bodies.

The derivation of the HM equations begins with the Newton and

Euler equations for an n-body system. Each of the two sets of equations

contains force terms representing the unknown constraint reactions which

occur at the joints between adjacent bodies. An expression for each

joint constraint force can be isolated by writing Newton's equations for

all the bodies that lie to one side of any selected joint. The equations

are added together, with the result that all the interaction forces can-

in pairs with the exception of the constraint force occurring at the

selected joint. Repeating the process for all the joints in the system

yields expressions for all the unknown constraints in terms of the system

external forces and in terms of the inertial accelerations of each body

in the system.

The joint constraint forces appear in Euler's equations as

torques about the individual body mass centers, and can be replaced by

the expressions For the constraints obtained in the process described.

The result is the original Euler equations for the system, but with the

unknown joint interactions explicitly eliminated. The result of the

7.

free to rotate (there may of course be more than just one) with the

summed vector dynamical equations for the bodies that lie to one side of

that joint. The value of the constraint torque can be computed after

'the modified HM equations are solved, however. Given the system state.

the derivatives of the state variables can be computed using the modi-

fied HM equations. All the variables in the equation which was dotted

with the joint degree of freedom vector ^ are then known, and can all be

brought to one side of the equation with the resulting sum equalling the

scalar components of the constraint torque (4:1207).

The interaction force at the joints can be computed by finding

the accleration of the mass center of the system of bodies lying to one

side of a joint (relative to the system mass center) and multiplying by*the total mass of that subsystem. For the case of no external forces,

the product is the vector constraint force acting at the selected joint.

Application of the HM Equations

The Eulerian-based equations of motion for multi-body systems

given by Hooker and Margulies (3) and modified by Hooker (4) are restric-

ted to those systems of bodies connected in such a way that no closed

loops are formed, and do not account for motion of the joints relative

to the bodies adjacent to the joint. This last restriction was removed

in an extenstion of the modified HM equations by Conway and Widhalm (5) to

permit the translation of the joint across the face of the OMV. Thus the

extended equations can be applied to the OMV, momentum wheels and target

satellite system under consideration. Referring to Fig. 2 the OMV is

labelled body 0, with the b basis fixed at its geometric center. The

geometric center of the OMV is chosen to coincide with the center of mass

- 8

of the OMV-momentum wheel composite. That is, remove the target from the

system and the system center of mass then lies at the origin of the

, frame. The momentum wheels lying on axes b1, 62, and 63 are labelled

bodies 1, 2, and 3 respectively. The target satellite is body 4. The

e basis is fixed in body 0 at its mass center, and the n basis is fixed

in the target at its mass center. Since the target is precessing about

its own angular momentum vector at a rate, b , the OMV is positioned

relative to the target so that the target's center of mass and angular

.2. momentum vector both lie on the b3 axis. The OMV is then spun about 63

at the same rate, ' the target precession rate. The cone angle, 4'LY4

and the distance from the target center of mass to the joint determine4the required position of the joint on the OMV face for docking. The

target's cone angle,)'4 , precession rate, , spin rate,Y 5 , and mass

properties are related as shown by Greenwood (6:386) and repeated here:

S1' = 5/(I 0 - I)cosV 4 (Y)

where I is the target's moment of inertia about the nI and n2 axes and

10 is the target's moment of inertia about the spin axis, A3 "

The two rotational degrees of freedom required at the universal

joint include rotation, Y4 , about an axis g4 parallel to b and rotation

Y5 9 about an axis g5 parallel to n3 . The rotational degrees of free-

dow for the momentum wheels labelled 1, 2, and 3 are axes il' 62' and i3-. respectively. These three axes are parallel to the corresponding 6 frame

axes, with the wheel joints themselves taken to be at the mass centers of

the wheels. From this initial docked configuration the problem is to

drive the system to a final spin-stabilized state with a set of feedba

. 94 1'

. .'

kb

controls. This final state is specified by requiring the joint location

to coincide with the 3 axis, and that the cone angle,Y , and spin rate

5 , be reduced to steady state values of zero.

In the following equations of motion, all vectors and scalar

rates are with respect to the e basis fixed in the main body, the OMV.

The attitude equations for the OMV-target satellite system can be derived

directly from the extended equations given by Conway and Widhalm (5) and

are:

a 0 a01 • • 05 0 "(mO40"m4140)

10 11 1' 1

- "21 g2 2 (2)

"3 3 E 3

a540 [ 4 + oD x G L ao 50• . .iJ 5J P5 • * t .+ mD4o x a .

where

a 0 u a dyadic00 :'k • , avector

aOk k p §k0 ~ ~(3)aio = gik OAP a vector

a gik gi " EiA(kl' " g , a scalar

1, if g belongs to a joint anywhere on the chain

and 4E. of bodies connecting p and the reference body (0)

0, otherwise (e.g. if iA= 0)

and04 x 04

10(e L

Oxx *0A + Ml 'k Ml (5)A]+~~ L~,D

*,=-m[ 6 lJ -DX\' (6)X= )mA (7)

DA CX A + (8)

The vector EA, is the vector from the center of mass of body A

to the joint leading to body p. is determined from

A = E kg k gk(9)

and Ex is the vector

yV xOA. @, -SDEA = 3YP OXx Ax. jJ Aj

+DA F+ Y. DA [F,; + M,,X (raX JA) (0

+ m Jl-3 ) .^DAP)l

In Eq (2) CO 0 I0e + U02e 2 +(J0 3e 3 and the Yi correspond

to degrees of freedom about the unit vectors gi. Superscript R implies

that the indicated time differentiation is performed with respect to an

observer fixed in the i-frame. From Eqs (7) and (8) and from the defi-

nition given for L X111 it is clear that the indicated time derivatives

of Df4 are determined from the universal joint motion alone since the

momentum wheel joints do not move with respect to the OMV. The joint

motion is specified as are the control torques -SD in Eq (10). In this

analysis, the external torques T' are the three orthogonal thruster

torques acting on the OMV, and all external forces are assumed zero. In

addition, all gravitational terms are ignored (all terms containing the

position vector ). The way in which the control torques (including

11

L ~~*A

thruster torques) and the joint motion is specified is covered in the

I next chapter.

. The constraint forces and torques have been eliminated from the

-' equations of motion as described earlier. These quantities can be deter-

mined by the methods described in references (3) and (4). The expres-

sions for the constraint force and torque at the universal joint are:

FH = (m - m4) 0 (11)044 P

T4' (044+ 040) •0 + (044 • 4) 4

(12)+ (044 * 95 5 7'- 1*4. - mU40 x

. where f0 is the position vector of the mass center of the OMV-momentum

wheel combination, relative to the system center of mass, and where U is

as given in Eq (4).

p

IV

j2

w'Q :$ "12

III. Nonlinear Feedback ControlIThe detumbling and despinning problem presented here involves

"In driving the universal joint connecting the 0MV and target satellites

across the face of the OMV to a point coincident with the b3 axis (see

Fig. 2). Feedback control is used to maintain the attitude of the five-

body system in such a way that the final state of the system is spin

stabilized. The Droblem solution incorporates a feedback control ap-

proach in which an eight element control vector, U, is a nonlinear func-

tion of the system state variables. In this chapter the eight controls

are defined so that the equations of motion can be written in a simpler

form more suitable for Liapunov analysis. This procedure closely paral-

lels that described by Widhalm and Conway (7:6-9) in their derivation of

a control law for the two-body satellite system described earlier.

Liapunov's direct method is then used to derive a control law which is

globally asymptotically stable with respect to the spin-stabilized equi-

librium state.

Eq (2) is first written in the form

A x = (13)

where A is defined as the 8 x 8 matrix on the left-hand side of Eq (2).

The vector, F*, is defined as the eight element vector on the right-hand

side of Eq (2), and

Tx = [k 1 2 x3 x4 X5 x6 x7 x8 ]

= [ "'01 "02 (3 'l '2 i3 V4 )5] (14)

13

Eq (14) is derived from the state variables

x= [ 1 x 2 x3 x4 x5 x6 x7 x8 ]T

C' 0l W02 '03 'l '2 3 4 5 ] (15)

The control vector, U, can be selected (7:6) so that Eq (13) can

be written as

A =F+ U (16)

Since the components of the vector, x, are the three angular acceleration

components of the OMV, the three scalar angular accelerations of the

othogonal momentum wheels, and the angular accelerations of the target

about the two degrees of freedom at the joint, the appropriate control

vector components are apparent. The control vector, U, is composed of

three orthogonal (thruster) torques about each of the i-basis vectors,

three internal torques applied at the wheel axes, and finally two inter-

nal torques applied in the two degrees of freedom of the universal joint.

These control torques are designated u1 through u8, respectively. Pre-

multiplying Eq (16) by the inverse of matrix A yields

x = A F + A U (17)

since the matrix, A, will always be invertible for physical systems. The

system of Eq (2) is augmented with the kinematical equation

k 9 = x7 (18)

where x9 is defined as the target precession angle, 74, and complates

* the set of equations of attitude motion. We define the augmented state

14

vector, iZ9, to contain the vector, 7, plus the ninth element, x9 .

To derive the control vector, U, as a nonlinear function of the

augmented state vector, 179 , and the joint motion, a lemma presented by

Vidyasagar (8) is applied. The lemma applies to autonomous systems, and

the system of Eq (16) is nonautonomous because the OMV-target connecting

joint motion is a specified function of time. Widhalm and Conway (7:7)

"* have suggested that, by specifying the joint motion as a third order lin-

ear system which is asymptotically stable with respect to the desired

- final joint position, the third order system with Eqs (17) and (18) form

an autonomous system. Since the desired final joint position relative

to the b-frame is given by the vector [ 0 0 c ] , and since the HM

equations are written in terms of the e-frame, we define the vector, Z,

to be the vector ledding from the origin of the b basis to the mass

center of the OMV, the origin of the e-frame. The scalar values that are

to be driven to zero for asymptotically stable joint motion are the posi-

tion, velocity, and acceleration components of the joint lying along the

62 axis. These scalars are given by:

( C+ L04) 62 = Yl

L4 b= 2= 'l (19)

L04~ 62 y3 =

The derivative of the vector, E, does not appear since the vector posi-

tion of the OMV mass center relative to the geometric center of the OMV

is constant. The complete autonomous system required for the application

of the previously mentioned lemma is given by Eqs (17), (18), and (19)

"• 15

~% N

and is formed by writing

x A- T AI -+_I

Xgx7 (20)9 7Dy7 D Y

where D is a negative definite matrix selected to obtain the desired de-

cay of the scalar yl given in Eq (19). The joint motion terms contained

in the matrix, A, and in the vector, F, are now specified by the vector

- leaving the system, (20), independent of time.

The lemma developed by Vidyasagar (8:157) in his discussion of

Liapunov's direct method now applies to the system, (20), and is stated

as follows: "Let V(ix 9. y) be continuously differentiable and suppose

that for some d > 0 the set

| * _

Sd = ix9 , 37: V(7q9 , Y)- d ]

is bounded. Suppose that V is bounded below over the set S* and thatdC

V(Ix 9 , Y) K 0 for all i79 and 7 in Sd*. Let S denote the subset of Sd de-

fined by

=S 1 [ 9, S Sd : V(li9 , 7) = 0 I

C-and let M be the largest invariant set of a system which is contained in

S. Then whenever iR9 (0) and Y(0) are members of Sd, the solution of the

system, (20), approaches M as t--."

Now M is an invariant set of system, (20), if every trajectory

starting from an initial point in M remains in M for all time. Since the

P" system, (20), is autonomous every trajectory through its state space isIL

16

55 :_ "" :; " ?i : -; " _ ,: : "?" ' ""? ::- ->"": ; *''- "": c

an invariant set (8:156). The task then is to find a candidate Liapunov

function, V, that meets the requirements of the lemma. To derive a non-

linear feedback control law that drives the five-body system to the de-

sired final state of spin-stabilized equilibrium, Widhalm and Conway (7)

suggested a candidate Liapunov function like

(1V (1/2) RT I 3 + (1/2) K x2 + yT R (21)

where I is the identity matrix, K is a positive constant, and R is a

positive definite constant matrix. The function is continuously differ-

entiable, and it is easy to select a vector R and a constant, d, to

demonstrate that a set like Sd exists.

The condition on V(- 9 , 7) must be satisfied; differentiating

V with respect to time yields

3J I3 T -X' + K x9 +9 y T + yT R - (22)

Substitution from the system, (20), gives the result

3 V Z3CT I ( A- ' "F + A-' U] + K x9 x7

+ yT [DT R + R D ] y (23)

Now since R is a positive definite matrix and D was specified to be a

negative definite matrix, then the expression DTR + RD is negative

definite. Writing the Liapunov matrix equation

-Q = DT R + R D (24)

and defining

_ T T (25)y Qy

17

implies both that Q is positive definite from Eq (24) and that as a

result both V and -V are positive definite (8:172). Thus the third

term in Eq (23) is negative definite. To make V at least negative sermii-

definite, select the control vector

5 =-F+A [ 0 0 0 0 0 0 -Kx9 0 T AB (26)

where the matrix B is positive definite or positive semi-definite. If

the matrix B is positive definite, then V is negative semi-definite in

x9. However, if the elements of B are positive except for B33 = 0 ,

V is negative semi-definite in x3 and x-. Then from the above lemma the

- system, (20), with control vector 9 of Eq (26) is asymptotically stable

with respect to the largest invariant set contained in the x3, x9 plane.

But from Eq (2b) it can be seen that any non-zero xg results in a non-

zero control vector U, which in turn causes a departure from the x3, x9

plane. Therefore the largest invariant set lies in the x3, x9 plane, but

. only in that region where x9 is zero, namely the x3 axis. The control

vector 0 of Eq (26) is then the desired nonlinear feedback control law

for spin-stabilization of this system.

Non-zero off-diagonal terms must be included in the B matrix of

Eq (26) which will couple the momentum wheel torques to state variables

other than just the momentum wheel angular velocities. The torques must

be coupled to the state variables in such a way that the angular rates of

h the gl and 92 wheels decay to zero, so that the system stabilizes in pure

spin. Implementing the control law of Eq (26) requires the determination

of the non-zero elements of the matrix B, and the constant K. In theirwork with the two-body satellite system, Widhalm and Conway have suggested

I

18

that controls can be kept within reasonable limits by ensuring that the

target center of mass does not depart appreciably from the b3 axis. This

can be accomplished by selection of the matrix D, which controls the uni-

versal joint motion, in conjunction with K and the B77 element of the B

matrix. Joint motion can be specified which will closely follow the de-

cay of the target's precession angle, and maintain the proximity of the

target mass center to the b3 spin axis. Finally, substitution of the

control vector 5 of Eq (26) into Eq (20) yields the complete, linear

system

i -Kx9 0T B

x9 = x7 (27)

y = D

From Eq (27) it can be clearly seen that if the matrix B is dia-

gonal, and if the initial momentum wheel angular rates as well as the

. OMV angular velocity components w01 and (0 are zero, they will remain

zero throughout the detumbling process. In the case where the only off-

diagonal terms in B are those coupling the momentum wheel torques to

selected system states, the cOl and ('02 angular velocity cohmponents of

the OMV still remain zero. This implies that the throughout the maneuver

the OMV remains in a state of pure spin.

4)..

19

IV. ResultsJIn this chapter the basis for the selection of initial conditions

K. and system mass properties is presented, as well as the values selected.

System state histories and control torque histories are presented for

four detumbling maneuvers. The four maneuvers include detumbling with

feedback control without momentum wheel coupling, feedback control with

a single momentum wheel coupled, detumbling using only gimbal (joint)

torques, and finally, detumbling using only gimbal torques for the first

250 seconds, followed by 50 seconds of feedback control applying the full

control torque vector with one wheel coupled. Representative histories

for the constraint loads at the joint between the OMV and the target sat-

ellite are also given.

System mass properties were selected based on values used in pre-

vious research (9). The mass properties of the composite body consisting

of the OMV and the three attached momentum wheels were selected to dupli-

cate those of the OMV model used by Widhalm (9). Reasonable but arbitrary

values of mass and moments of inertia were then selected for the three

identical wheels. From this information and by specifying the location

of the mass center of the composite OMV-momentum wheel system, the mass

and inertia matrix for the OMV (without wheels) were computed. In this

way direct comparisons could be made between the two-body and five-body

analyses by completely decoupling the wheel control torques from all non-

zero system states. As mentioned previously, this is accomplished by

selecting all off-diagonal terms of the matrix B in the linear system

(27) to be zero. Under this condition, the values of elements B44, B55,

and B66 are completely arbitrary. In addition to the various mass

.66

20

.1

properties, the system state variable initial conditions were also dupli-

cated from reference (9). The mass properties and initial conditions are

given in Tables I and II, respectively.

From Eq (27) and from the discussion that immediately follows it,

it can be seen that the values selected for elements B11 and B22 are also

arbitrary, since they will always be multiplied by states x and x2, which

, remain equal to zero throughout the detumbling, despinning maneuver. The

values actually selected are those used by Widhalm (9) in the application

* of feedback control to the two-body satellite system. In that case states

x and x2 attained non-zero values after approximately 290 seconds of

open-loop control. Feedback control was then applied in an attempt to

spin-stabilize the system, the desired response being obtained using the

values B11 = B22 = 0.046

The remaining constants to be determined include B77' B88 , K, and

the elements of the D matrix of the third order joint motion equation in

Eq (27). Selecting the scalar equation from the 12 equation system of

Eq (27) corresponding to the target precession angle motion, the constants

- B77 and K can be determined by specifying a desired final precession

* angle at the end of the maneuver. A total maneuver time of 300 seconds

was selected, and critical damping specified. A final precession angle

corresponding to the target mass center on the OMV spin axis and a final

joint position equal to 0.05% of initial joint position was specified.

The result obtained from these requirements is a solution with equal

eigenvalues of about -0.035, with B77 = 0.07 and K = 0.00123. In a

similar way, with the specification of the final joint position just

given and with critical damping of the joint motion, equal eigenvalues

21

TABLE I

Satellite Mass Properties

MASS I1 12 13

Target Satellite 1000 Kg 1000 Kg-m2 1000 Kg-m2 1100 Kg-m 2

0MV 4500 Kg 6400 Kg-m2 6400 Kg-m 2 11800 Kg-m 2

Momentum Wheels 10 Kg 25 Kg-m2 25 Kg-m2 55 Kg-m 2

TABLE II

Initial Conditions

Variable Meaning Value

(. 01 e1 component OMV angular velocity 0.0

(J02 a2 component OMV angular velocity 0.0

W 03 e3 component OMV angular velocity 0.102

1 #1 momentum wheel ;pin rate 0.0

2 #2 momentum wheel spin rate 0.0'3 #3 momentum wheel spin rate 0.0

4 target precession angle rate 0.0

"15 target spin rate 0.009

(L04 + ). b2 joint position 0.599

L b joint velocity 0.0Ln4 2

L04 b 2 joint acceleration 0.0

>14 target precession anqle 0.349

Note: Values in TABLE II are given in meters, radians,or radians per second, as applicable.

p.222

U

of -0.04 were obtained for the joint motion equation. This resulted in

values of D31 = -0.000064, D32 = -0.0048, and D33= -0.12 . Finally,

solving the eighth scalar equation of the system of Eq (27) by requiring

that the target spin rate be reduced to 0.5% of its initial value yields

the value of B88 = 0.02.

With the momentum wheels uncoupled the system behavior was ident-

ical to that of the two-body system in Fig. 1, verified by the comparison

of the results with those of Widhalm (9). Wheel torques were zero, with

the OMV thrust torques and universal joint torques given in Figs. 3 and 4.

None of these control torques reached large values, and it can be seen

4 that the controls vary smoothly with time with no abrupt changes. The

joint motion and precession angle decay behaved as specified, decaying

to the small final values specified with no overshoot. The results are

given in Fig. 5. The target spin rate decay displayed similar behavior.

The precession angle rate of change and target spin rate histories are

given in Fig. 6, where it can be seen that no radical motion has occurred.

* The constraint loads at the universal joint due to lack of rotational

freedom and due to the loading required to drive the joint across the OMV

were relatively small. The constraint torque maximum value was approxi-

mately 2 Nm, and the maximum force encountered in driving the joint was

about 1 N. These results are shown in Fig. 7, where the magnitude of the

constraint force is plotted, as well as the component of that force lying

ZI along the axis of joint motion. This b2 component corresponds to the load

7 'capacity that would be required for a jackscrew, for instance, were such

a device used on the OMV to obtain the desired joint motion.

Although global asymptotic stability is only guaranteed by the

23

I

4,

lemma used earlier for the case of feedback control with the control law

of E(26), it was of interest to attempt system despin and detumbling

using only control torques applied at the universal joint. The result

was that the system displayed no radical or unusual behavior, and appro-

ached a nearly spin-stabilized state after 300 seconds of control. The

OMV still remained in a state closely approximating pure spin, although

.4 non-zero angular velocity components did remain. Figs. 8 and 9 show the

OMV angular velocity history for this maneuver, and indicate that the de-

parture from pure spin was modest. Constraint torque and interaction

force at the connection also remained small, although as indicated in

Figs. 10 and 11, a quasi-steady state condition was reached with continu-

ous constraint and control torques experienced at the joint. A residual

precession angle remained after the maneuver, but the rate of change of

the precession angle was very small, indicating that the available con-

trol torques were able to maintain the system configuration but unable to

change it at any appreciable rate. Although there was some oscillation

in the precession angle rate of change, no other violent behavior occur-

ed with respect to either target spin or precession angle changes. The

results are shown in Figs. 12 and 13.

" -The maneuver described above was repeated, but at 250 seconds the

full control vector was applied in an attempt to spin-stabilize the sys-

tem. The residual non-zero c02 component of the OMV angular velocity

was reduced nearly to zero, and the OMV spin rate brought to a steady

value (see Figs. 14 and 15). The relative spin rate of the target and the

* precession angle rate of change were likewise reduced to very nearly zero

as indicated in Fig. 16. Although not plotted, the precession angle of

24

fV.

UI

L2

the target was reduced to half of its value at 250 seconds, or approxi-

mately 0.9 degrees. Figs. 17 through 19 show the attendant reductions

- .~.in the constraint and control torques and in the constraint force at the

joint. These data clearly indicate that the system is moving toward a

pspin stabilized equilibrium.

An attempt was made to couple the momentum wheels to the target

precession angle rate and spin rate. The magnitude of the OMV thruster

torque about each of the b axes was integrated over the maneuver period,

and the three resulting values added to obtain some measure of the total

torque power required from the thrusters. The resulting total was then

compared for maneuvers completed using various off-diagonal gain terms inIA

the B matrix of Eq (26). No attempt at using control torque on the b3

wheel yielded a reduction in the OMV thruster torque requirements. Using

applied torque at the b2 wheel by setting B57 to 1.0 resulted in a reduc-

tion of the integrated thrust torque from 436 Nm-sec for no wheel coupling

,. to approximately 377 Nm-sec. The resulting thruster control torques and

momentum wheel control torque are shown in Figs. 20 and 21. Fig. 21 also

includes the b2 momentum wheel angular rate history during the maneuver.

These results indicate that the system is well behaved in the

sense that despinning and detumbling can be accomplished with relatively

small control requirements, and that no violent dynamical behavior occurs

even when internal control torques only are applied. Also, the use of.4 .

momentum wheels to reduce thruster requirements is possible, although no

evidence is presented that the wheels provide any major benefit in terms

of efficient system control.

N 25

C 4.0-Ul

2.0 -

a-0-0 -

0.0 50.0 100.0 150.0 200.0 250.0 300.04 TIME, SEC

Fig. 3. OMY Thrust Torques, Momentum wheels Uncoupled.

2. 0

U8

;10-0 - ---------------------------------

-2.0-10.0 50.0 100.0 150.0 200.0 250.0 300.0

TIME, SEC

4 Fig. 4. Gimbal (Universal Joint) Torques, Uncoupled Momentum Wheels.

26

1! .00 -_ _ _ _ _ _ _ _ _ _ _ _ _ _

S- Precess. Ang., rad

-------- Joint Position, m0.75

0.50 "%

0.25-

0.00 -0.0 50.0 100.0 150.0 200.0 250.0 3-00.0

TIME, SEC

Fig. 5. Joint Position and Precession Angle Decay

0.010------ Target spin, rad/sec

____Precess. Ang. rate,1.0 rad/sec

0-005-'

0.000-

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 6. Target Spin and Precession Rate Decays

27...

'li !--0. 05 -I I 27

i} 0. 00£O . 5 .02 00.5 . 0 .

TIEIE

i ig 6. Tage pi ndPecsio t ecy

-.-:- 2,0-

1-- 0.0 - --

---- IFC1, N

-2.0

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 7. Constraint Loads at Universal Joint

2

I!

4

28

'I

,U

,.,.- , ,,w - .,v L , - ,:,,,,:,: ' "4'i - , . .p "- . 2 ." .'.' . ....... .-. . .-.-- .". . - . . ;.. - . .-.;

0.02

:".0 01 - -----------------------

J/

/

0.00-- ,,rad/secoa, rad/sec

-0.01-0.0 50.0 100.0 150.0 200.0 250.0 300.0

TIME, SEC

". Fig. 8. Angular Velocities of OMV with u7 and u8 Feedback Only-?*

0.106-* .°9

0.104- C ,, rad/sec

~0-102- -°

4i 0.100

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 9. Angular Velocity ca3of OMV with u7 and u8 Feedback Only

29

k.

4-0-

2.0 -- F FCI N

II

-2.0-F C N2'

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

*-Fig. 10. Joint Constraint Loads, u7and u 8 Feedback Only

00- 0 -----

u 8 torque, Nm

- u7 torque, Nm

-~ -2.0-

-4.0-

-4.* .. " ,

0.0 50.0 100.0 150.0 200.0 250.0 300.0

me L

L' TIME, SEC

'"Fig. 11. Cointo osruaitLoads, u 7 and u 8 Feedback Only

\,.3-'" 30

0.010 Target spin, rad/sec

-----Precess. Angle Rate of

Change, rad/sec

0.005-

0.000- --------------------

-0.005

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME,SEC

Fig. 12. Target Spin and Precession Angle Rates, u7 and u, Feedback Only

0 6- %- Precess. Ang., rad

- Joint Position, r

0.4-

0.2 '

000 50.0 100.0 150. 200. 250.0 300.0TIME, SEC

Fig. 13. Target Precession Angle and Joint Position, u 7 and u 8Feedback Only

31

4,, ,

0.015

0.010- .

"0-005-'

0.000-...:/ Wo,, rad/sec

-0.005- @ , rad/sec

-0.010

0.0 50.0 0O0.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 14. OMV Angular Velocity Components; Full Feedback Addedat t 250 seconds

0.102-(a.., rad/sec

0.101-

S0.lO 0 00.0 50.0 100.0 150.0 200.0 250.0 300.0

TIME, SEC

Fig. 15. OMV b3 Angular Velocity Component; Full Feedback Addedat t 250 seconds

32

6*

0.010

Precess. Angle Rate ofU Change, rad/sec

0.005- - Spin rate, rad/sec

0.000-oo o --------------

-0.005-

-0.010 -

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 16. Precession Angle Rate of Charge and Target Spin Rate; FullFeedback Added at t 250 seconds

0.0-Torque in Newton-meters

-4.0-

3,3

-6.0

0.0 50.0 100.0 1,50.0 200.0 250.0 300.0' • TIME, SEC

Fig. 17. Gimbal Control Torque u7 With Full Feedback Added[i at t = 250 seconds

"" 33

q 2S'&

1.0

* 0.0

it,,

NmT Nm,

-2.0 "

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 18. Control Torque u8 and Constraint Torque TC; Full Feedbackat t = 250 seconds

4.0-

F 2.0-0 -I

0.0.

-2.0 ---.

-4.0

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 19. Constraint Force Magnitude and 6 Component; FullFeedback Added at t 250 secongs

34

oIlk,

0.0 -

/Js,, Nm

u Nm/# 2

- .0 - 3

• ., : -4.0 ,,

0.0 50.0 100.0 150.0 200.0 Z50.0 300.0TIME, SEC

Fig. 20. OMV Thruster Torques, b Momentum Wheel TorqueCoupled to Target Preceision Angle

0.3..,, u5 Nm

:, , rad/sec

0.2P --

0.1 /

0.0

-0.1

0.0 50.0 100.0 150.0 200.0 250.0 300.0TIME, SEC

Fig. 21. b Momentum Wheel Control Torque, u , and Wheel Angular2 Vel.; Control Torque u5 Coupled o Precession Angle

.,4 > 35

I

TABLE III.

time u u8 TC IFCI F2

0 -1.500 -. 192 .000 .613 -. 5765 -.822 -. 242 -.984 .503 -.225

10 -. 176 -.267 -1.576 .631 .09715 .399 -.274 --1.897 .759 .37420 .884 -. 267 -2.032 .859 .60125 1.273 -.251 -2.043 .935 .77830 1,568 -.230 -1.973 .993 .90735 1.777 -.207 -1.853 1.033 .99240 1.909 -. 183 -1.705 1.056 1.04145 1.976 -. 161 -1.545 1.061 1.05850 1.989 -. 140 -1.382 1.051 1.05055 1.959 -.121 -1,224 1.027 1.02360 1.895 -. 104 -1.076 .990 .98165 1.808 -.090 -.939 .945 .92870 1.703 -.077 -.815 .892 .86975 1.588 -.067 -.704 .835 .80680 1.468 -.058 -.606 .775 .74185 1.346 -.050 -.520 .714 .67690 1.226 --.043 -.445 .654 .61395 1.109 -.038 -.380 .595 .553100 .998 -.033 -.324 .538 .496105 .894 -.029 -. 276 .484 .443110 .797 -.025 -. 235 .434 .393115 .708 -.022 -.200 .387 .348120 .626 -. 020 -. 170 .344 .307125 .552 -.018 -. 144 .304 .270130 .485 -.016 -. 123 .268 .236135 .425 -.014 -. 104 .236 .207140 .371 -. 012 -. 089 .207 .180145 .323 -.011 -.076 .181 .156150 .281 -. 010 -. 064 .157 .136155 .244 -.009 -. 055 .137 .117160 .211 -. 008 -. 047 .119 .101165 .182 -.007 -.040 .103 .087170 .157 -.007 -.034 .089 .075175 .135 -.006 -.029 .077 .064180 .116 -. 005 -. 025 .066 .055185 .100 -.005 -.022 .057 .047190 .085 -. 004 -. 019 .049 .040195 .073 -.004 -.016 .042 .034200 .062 -. 004 -. 014 .035 .029205 .053 -.003 -.012 .030 .025210 .045 -. 003 --.010 .026 .021215 .038 -.003 -.009 .022 .018220 .033 --.00.2 -. 008 .019 .015225 .028 -. 002 -. 007 .016 .013230 .023 --.002 -,006 .013 .011

36

iAl

TABLE III., cont'd

time u7 u8 TC IFCI FC

235 .020 -.002 -'.005 .011 .009240 .017 -.002 -.004 .010 .008245 .014 -.001 -.004 .008 .006250 .012 -.001 -.003 .007 .005255 .010 -.001 -,003 o006 .005260 .008 -.001 -.002 .005 .004.265 ,007 -.001 --.002 ,004 .003270 o006 -0001 -.002 .003 .003275 .005 -.001 -.002 .003 .002280 .004 -1001 -.001 .002 .002285 .004 -.001 -.001 .002 .002290 .003 -.001 -.001 .002 .001295 .002 -.001 -.001 .001 .001

Note: This table of universal joint control torques andconstraint loads applies to the case of feedbackcontrol with uncoupled momentum wheels.

,i3

,1i

4b

-V 37

6

TABLE IV.

time u7 u8 TC IFCI Fc7 8 2

0 -1.500 -. 192 .000 *613 -. 5765 -.822 -.242 -.984 .503 -.225

10 -. 176 -. 267 -1.576 .631 .09715 .399 -.274 -1.897 .759 .37420 .884 -. 267 -2.032 .859 .60125 1.273 -. 251 -2.043 .935 .77830 1.568 -.230 -1.973 .993 .90735 1.777 -.207 -1.853 1.033 .99240 1.909 -. 183 -1.705 1,056 1.04145 1,976 '-#161 -1.545 1.061 1,05850 1.989 -. 140 -1.382 1051 1.05055 1.959 -. 121 -1.224 1027 1.02360 1.895 -. 104 -1.076 .990 .98165 1.808 -.090 -.939 .945 .92870 1.703 -. 077 -.815 .892 .86975 1.588 -.067 -.704 .835 .80680 1.468 -.058 -.606 .775 .74185 1.346 -.050 -.520 .714 .676

190 1.226 -. 043 -.445 .654 .613195 1109 -.038 -.380 .595 .553100 .998 -.033 -.324 .53 .496105 .894 -.029 -.276 .484 .443110 .797 -. 025 -.235 .434 .393115 .708 -. 022 -.200 .387 .348120 .626 -.020 -. 170 .344 .307125 .552 -.018 -. 144 .304 .270130 .485 -.016 -. 123 .268 .236135 .425 -.014 -. 104 .236 .207140 .371 -.012 -.039 .207 .10145 .323 -.011 -. 076 .181 .156

* 150 .281 -.010 -.064 .157 .136155 .244 -.009 -.055 .137 .117

* 160 .28 -.008 -.047 .119 .101" 165 .182 -.007 -.040 .103 .087

170 .157 -.007 -.034 .039 .075175 .135 -.006 -.029 .0077 .064180 .116 -. 005 -.025 .066 .055185 .100 -. 005 -.022 .057 .047190 .085 -.004 -.0019 .049 .040

' 195 .073 -. 004 -,016 .042 .034... 200 .062 -.004 -,014 .035 .029

205 .053 -*003 -.012 .030 4,025" 210 .045 -6003 -.010 .026 .021hr 215 .038 -. 003 -0009 .022 *018

220 .033 -.002 -. 008 .019 .015225 .023 -.002 -.007 .016 .013

+230 .023 --.002 -.006 .013 6011

38

I:

TABLE IV. cont'dtime u u TC IFCI FC

tie7 8 2

235 .020 -.002 -.005 .011 .009240 .017 -.002 -.004 .010 .008245 .014 -.001 -.004 .008 .006250 .012 -.001 -.003 .007 .005255 .010 -.001 -.003 -006 .005260 .008 -.001 -.002 *005 .004265 .007 -.001 -.002 .004 .003270 .006 -.001 -,002 .003 .003275 .005 -.001 -.002 .003 o002280 .004 -.001 -.001 .002 .002285 .004 -.001 -.001 .002 .002290 .003 -0001 -.001 .002 .001295 .002 -.001 -.001 .001 .001??

Note: This table of universal joint control torques andconstraint loads applies to the case of feedbackcontrol with the b2 momentum whe2l coupled.

P!3

* 4

.4V 39

TABLE V.

U time u1 u2 u3 u5 2

0 -1.732 -.066 .180 .000 .0005 --.960 -1.209 .182 .096 .005

10 -. 242 -1.906 .205 .157 .01615 .382 -2,296 ,238 .191 .03220 .893 -2.473 .269 .206 .05125 1.286 -2,504 .291 .207 .06930 1.566 -2,438 .301 .199 .08835 1.744 -2.311 .299 .184 .10540 1.832 -2.147 .287 .165 .12145 1,845 -1.964 ,268 .145 .13550 1,799 -1.777 .244 .123 .14855 1,706 -1.593 .218 .102 .15860 1.579 -1.418 *192 .082 .16665 1,430 -1.256 ,167 .062 .17370 1.267 -1108 o144 .045 .17875 1.098 -.974 *122 .028 .18180 .929 -.855 ,104 .014 .18385 .764 -,750 ,088 ,.001 .183

90 .607 -. 657 .074 -.010 .18395 .461 -.576 .062 -.019 .182100 .325 -o506 .052 -.028 .180105 .203 -,445 .044 -.035 .177110 .093 -. 392 .037 -.040 .173115 -:005 -:346 .032 -. 045 .169

S120 -. 090 -. 306 ,027 -,048 ,165

125 -,163 -.271 .023 -.051 .161-130 -.226 -,242 *020 -.053 .156135 -.278 -.216 .017 -.055 .151140 -. 322 -. 194 *015 -.056 .146145 -. 357 -. 174 .013 -.056 .141150 -.385 -. 157 .011 -.056 .136155 -.406 -.142 0010 -.056 .131160 -. 422 -,129 ,009 -.055 .126165 -.433 -. 118 .008 -.054 .121170 -.439 -. 108 ,007 -.053 .116175 -,441 -. 099 ,006 -.052 .11.1180 -.441 -.091 .006 -,051 .106185 -. 437 -.084 .005 -. 049 .102190 -. 432 -.077 .004 -. 048 .097195 -. 424 -. 072 .004 -.046 .093200 -.416 -.067 .004 -. 045 .089205 -.405 -,062 .003 -.043 .085210 -o394 -.058 .003 -. 041 081215 -. 383 -.054 .003 -,040 .077220 -.371 -o050 .002 -.038 .074225 -.358 -,047 ,002 -.037 .070230 -.345 -,044 .002 -,035 .067

40

4o

*. TABLE V. cont'd

time u1 u2 u3 u5 2

235 -.333 -.041 .002 -.034 .064240 -.320 -.039 .002 -.032 .061245 -.307 -.037 .001 -.031 .058250 -.295 -.035 .001 -.030 .055255 -.283 -.033 .001 -.028 .053260 -.271 -.031 .001 -.027 .050265 -.259 -.029 .001 -.026 .048270 -.248 -.027 .001 -.025 .046275 -.237 -.026 .001 -.023 .043280 -.226 -.024 .001 -.022 .041285 -.216 -.023 .001 -.021 .039290 -.206 -.022 .001 -.020 .037295 -.197 -.021 .001 -..019 .036

Note: This table of thruster torques u , u., and u andwheel torque u5 applies to the c lse bf feedbickcontrol with the b2 momentum wheel coupled.

A4

Sr.

K,41

I;

A4

TABLE VI.

Utime T~ I FC I F c7 8 2

0 -1,500 -*192 .021 .612 -.5005 -.830 -,281 -.471 .340 -.225

10 -. 403 -*314 -.738 .242 -.09415 -.201 -. 298 -.882 .223 -.071

20-.180 -.251 -.974 .237 -. 12425 -.285 -.192 -1,054 .288 -,22330 -.463 -. 136 -1.139 .379 -.346

35-.671 -.093 -1.231 .495 -647640 -0878 -.068 -1.323 .619 -.60245 -1.067 -o063 -1.402 .740 -t72150 -1.230 -.074 -1.454 .851 -.83155 -1.371 -.095 -1.470 .953 -,93660 -1.499 -.119 -1.444 1.051 -1,03865 -1.629 -. 139 -1.379 1.150 -1,14170 -1.771 -.151 -1.280 1.253 -1,24875 -1.933 -.151 -1.159 1.363 -1.35980 -2.119 -.141 -1.029 1.478 -1*47385 -2.322 -. 122 -.900 1.596 -1.58990 -2,537 -*099 -.784 1.712 -1,70495 -2.751 -.075 -.687 1.822 -1,813100 -2.954 -.054 -.612 1.922 -1,914105 -3.139 -.036 -.559 2.012 -2,003110 -3.300 -.024 -.524 2.088 -2.080115 -3.433 -.016 -.504 2.152 -2.143120 -3.541 -.011 -.492 2.204 -2.194125 -3.626 -.1010 -.483 2.245 -2.235130 -3.693 -.010 -.473 2.279 -2.268135 -31.746 -.010 -.458 2.306 -2.295140 -3o792 -.010 -.440 2.330 -2.318I145 -3.834 -.009 -.416 2.351 -2.338150 -3.873 -.008 -*391 2.371 -2.357155 -3.912 -,005 -.364 2.389 -2,375160 -3.950 -.003 -.340 2.407 -2.392165 -3,986 .000 -.318 2.423 -2.408170 -4,020 .003 -.301 2,437 -2.422175 -4.049 .005 -.288 2.449 -2.434180 -4.072 .006 -.280 2.45B -2.443185 -4.090 .007 -.276 2.465 -2.449190 -4.103 .007 -.274 2.469 -2.453195 -4.110 .006 -.273 2.471 -2.455200 -4.113 .006 -.272 2.471 -2.455205 -4.114 .005 -.271 2.470 -2.454210 -4.112 .004 -.269 2.469 -2.452215 -4.110 .003 -,266 2.466 -2.450220 -4.107 .003 -.262 2.464 -2,447225 -4.104 .003 -.258 2,461 -2.445230 -4,101 .003 -. 253 2,459 -2.442

42

TABLE VI. cont'd

time u u8 TC IFCI FC7 2

235 -4.099 .003 -.249 2.456 -2.440

240 -4.096 .003 -.246 2.453 -2.437245 -4.093 .003 -.243 2.450 -2.434250 -3.847 .003 -2.162 2.495 --2.284255 -3.192 .003 -1.813 2.072 -1.896260 -2.643 .004 -1.519 1.720 -1.572265 -2.194 .005 -1,276 1.432 -1.306270 -1.824 .005 -1.075 1.194 -1.087275 -1.519 .006 -.907 .998 -.906280 -1.267 .006 -.768 .836 -.757285 -1.059 .006 -.651 .702 -.634290 -.887 .006 -.553 .590 -.532295 -.743 .006 -.471 .497 -.446

Note: This table of universal joint control torques andconstraint loads applies to the case of feedbackwith control torques u7 and u only until t = 250seconds, at which time the complete control vectoru is fed back.

43

m t

TABLE VII.

time '01 W02 W 03 4

0 .000 .000 .102 .000 .0095 .001 .001 .102 -. 003 .008

10 .001 .002 .102 -.004 .00615 •000 .003 .102 -.004 .00520 -.001 .004 .102 -.003 .00425 -. 002 .004 .101 -.002 .00330 -. 004 .004 .101 -. 001 .00335 -. 005 .004 .101 .000 .00240 -.005 .003 .101 .000 .0045 .-.005 .002 .101 .000 .00250 -.005 .002 .101 -.001 .00155 -004 .001 .101 .-.002 .001

• 60 -. 003 .002 .101 -.003 .00165 -. 002 .002 .101 -.004 .00170 -.001 .003 .101 -. 004 .00075 .000 .004 .101 -.005 .000

0 80 .000 .005 .101 -. 004 -. 00185 .000 .006 .101 -. 004 -.00190 .000 .007 .101 -.004 -.00195 .000 .008 .101 -. 003 -. 001

100 -.001 .008 .101 -.002 --.002105 -.001 .009 .101 -.002 -. 002110 -.-#001 .009 .101 -. 001 -. 002115 -.001 .009 1101 -.001 -.002120 -.001 .009 .101 -. 001 -. 002

130 -.001 .009 .101 -. 001 -.002135 -.001 .009 .101 -. 001 -.002140 -.001 .009 .101 -. 001 -. 002

145 -.001 .009 .101 -.001 '-.002150 -. 001 .010 .101 -. 001 -. 002155 .000 .010 .101 -.001 -.002160 .000 .010 .101 -. 001 -.002

165 -.001 .010 .101 -. 001 -.001170 -. 001 .010 .101 .000 -. 001175 -.001 .010 .101 .000 .-.001

, 180 -. 001 .010 .101 .000 -. 001185 -.001 .010 .101 .000 -. 0011 -.001 .010 .101 .000 -.001195 -.001 .010 .101 .000 -.001200 -. 001 .010 .101 .000 -. 001

205 -.001 .010 .101 .000 -.001210 -.001 .010 .101 .000 -.001

C- 215 -.001 .010 .101 .000 -f001220 -. 001 .010 .101 .000 -. 001225 -1001 .010 .101 .000 -.001230 -. 001 .010 .101 .000 -. 001

44

TABLE VII. cont'd

time 01 02Y4235 00 .103 y5

235 -.001 .010 .101 .000 -.001' 240 '-.001 .010 .101 .000 -.001

' 245 -. 001 .010 .101 .000 -. 001

250 -.001 ,009 .101 .000 -.001255 .000 .008 .101 .000 -.001260 .000 .006 .101 1000 -.001265 .000 .005 .101 .000 -,001270 .000 .004 .101 .000 -.001275 .000 *003 .101 .000 -.001280 .000 .002 .101 .000 .000285 .000 .002 .101 .000 .000290 .000 .002 .101 .000 .000295 .000 .001 .101 .000 .000

NOTE: This table of OMV angular velocity components,target spin rate and precession angle rate ofchange is for the case of feedback with u7 andU 8 control vector components only, until

t = 250 seconds when control is with thecomplete u vector.

,45

V. Conclusion

A nonlinear feedback control law was developed and used to despin

and detumble an axially symmetric target satellite originally in steady

spin and precession. The control law derivation is based upon Liapunov

stability theory, and ensures the global asymptotic stability of the

final spin-stabilized equilibrium state. The results indicate that the

system is well behaved, in the sense that changes in both the system

state and in the control torques are smooth throughout the maneuver. The

control torque magnitudes are relatively small, and no extreme loading of

the connecting joint between OMV and target satellites occurred. The

system could be driven very close to the spin-stabilized state using the

joint control torques alone. However, a residual target precession angle

remained at the end of the 300 second maneuver, as did non-zero bI and b2

OMV velocity components. This is due to the fact that after approximately

200 seconds of feedback control, the matrix, A, of Eq (26) is nearly dia-

gonal, and as a result the control torques u7 and u8 are coupled strongly

only to the target precession angle rate and spin rate, and the precession

angle (states x7, x8, and x9), all of which have very small values. The

" available control torques at the universal joint are thus insufficient for

the reduction of target precession angle at any appreciable rate. Imple-

mentation of full vector control (using all eight control torques) at

t = 250 seconds successfully drove the system to the spin-stabilized

equilibrium.

Although the OtlY thrust torque magnitude could be reduced by

coupling a momentum wheel torque to the target precession angle, no at-

tempt was made to accomplish detumbling with momentum wneel and joint

46

torques alone. Any attempt made to couple the b3 momentum wheel with

system states x7 or x8 resulted in an increase in at least one thrust

torque profile, with no obvious positive influence on system behavior.

A follow on effort might concentrate on the development of a

reliable technique for determining the values of the off-diagonal gain

matrix terms, based on desired system response. Then an attempt could

be made to perform the bulk of the maneuver using only internal torques.

i.4

47-

References

1. Conway, B.A., and Widhalm, J.W., "Optimal Continuous Control forRemote Orbital Capture," AIAA J. of Guidance, Control, and Dynamics,to appear.

2. Fletcher, H.J., Rongved, L., and Yu, E.Y., "Dynamics Analysis of aTwo-Body Gravitationally Oriented Satellite," Bell System Tech. J.,Vol. 42, No. 5 (1963) 2239-2266.

3. Hooker, W.W., and Margulies, G., "The Dynamical Attitude Equationsfor an n-Body Satellite," J. of the Astronautical Scietices, Vol. 12,No. 4 (Winter, 1965) 123-178.

4. Hooker, W.W, "A Set of r Dynamical Attitude Equations for an Arbitraryn-Body Satellite Having r Rotational Degrees of Freedom," AIAA J.,Vol. 8, No. 7 (1970) 1205-1207.

5. Conway, B.A., and Widhalm, J.W., "Equations of Attitude Motion for ann-Body Satellite With Moving Joints," AIAA J. of Guidance, Control,and Dynamics, Vol. 8, No.4 (1985) 537-33 -_g.

6. Greenwood, D.T., Principles of Dynamics. Prentice-Hall, Inc.,Englewood CliffsNJ,1965.

7. Widhalm, J.W., and Conway, B.A., "Nonlinear Feedback Control forRemote Orbital Capture," paper No. AAS85-368 presented at AAS/AIAAAstrodynamics Specialist Conference. AAS Publications Office,PO Box 28130, San Diego, CA (August 1985).

8. Vidyasagar, M., Nonlinear Systems Analysis. Prentice-Hall, Inc.,Englewood Cliffs,1lFTM._

9. Widhalm, J.W., "Optimal Open Loop and Nonlinear Feedback Control forRemote Orbital Capture,"1 1.D7Thess, University of Ilinois at-rbana-champaign,--T85.

48

, . .. ... , , * N , , A •. A,

S.

VITA

Lieutenant Kirk R. Fleming was born on 13 October 1954 in Port

Huron, Michigan. He graduated from Port Huron Northern High School in

January 1972, and attended the University of Texas at Austin from Aug-

ust 1979 to December 1981, when he received the degree of Bachelor of

Science in Aerospace Engineering. He received a commission in the

USAF through the Air Force Officer Training School in April 1982. He

then served as a research and development project engineer in the Air

Force Flight Dynamics Laboratory, Vehicle Equipment Division, Wright-

Patterson AFB, Ohio until entering the School of Engineering, Air Force

Institute of Technology, in May 1984.

Permanent address :1954 Burns Road

Smith's Creek, MI 48074

4-

,9

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FIELD GROUP SUB.OR. satellite, dynamics, teleoperator, orbital capture22 02docking, detumbling, despinning, feedback control

19. ABSTRACT (Continue on reverse if neceseary and identify by block number)

Title: THE DETUMBLING OF AN AXIALLY SYMMETRIC SATELLITE WITH AN

ORBITAL MANEUVERING VEHICLE BY NONLINEAR FEEDBACK CONTROL

Theis dvior:Lt. Co.Joseph .ihl

Deam01 1011 Aeear0cI and Prf ind I. beg 9nAir Weeom Iammge @Ifcat (~A M-WdOW410ftom &" oil 4M

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From Block 19:

The problem of detumbling a freely spinning and precessing 1axially symmetric satellite is considered. Detumbling is achieved jwith another axisymmetric orbital maneuvering vehicle (OMV) joined

to the target satellite with a universal joint. The joint provides

two rotational degrees of freedom and is translated across the sur-

face of the OMV during the detumbling process. The target satellite

and the OMV with its three momentum wheels are modelled as a five

body system using Eulerian-based equations of motion developed by

* Hooker and Margulies. A Liapunov technique is applied to derive a

nonlinear feedback control law which drives the system asymptotically

to a final spin-stabilized state. State and control histories are

presented and indicate that the detumbling process is benign. Con-

straint force and moment loads at the connection between the OMV and

target satellites are also presented, and indicate that no extreme

loads are encountered during the debpinning and detumbling process.

S!

! ?.

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